All traits were treated as normally distributed and analyzed incorporating fixed effects and covariates based on the models reported by Valdar et al. . Fixed effects were sex (W6, W6m, W10, TA, FB, HC, I75), year-month (W6, W6m, W10, TA), parity (W6, I75), experimenter (TA, I75), apparatus (TF) and month (I75); covariates comprised cage density (W6, W6m, W10, I75), age in days (W6, W6m, W10) and weight (HC, I75). Cage was added as a random effect for all traits. Cages consisted almost solely of animals from one family. For all practical purposes cage was nested within family (avg. 3.1 cages per family).
Three basic groups of models were used to compare changes in variance components and PA as a result of using genomic information. One model used only polygenic effects (1), a second model used only genomic effects (2), and a third model fitted both effects (3). For models (2) and (3), seven different sub-models were considered based on the percentage of markers that was assumed to have a substantial effect. This included a non-mixture model using 100% and six mixture models, ranging from 70%, 40%, 10%, 7.5%, 5% to 2.5% of the SNPs assumed to have a substantial effect. In the following, these sub-models will be labelled according to the mixture percentages. All analyses were performed using a Bayesian approach and implemented with Markov chain Monte Carlo methods [30
] using the programme iBay [31
]. The basic model using polygenic effects can be described as follows:
fits a general mean and the vectors b, c, u
and residual effects
, respectively. I
is the identity matrix and A
the additive genetic relationship matrix. X
are incidence matrices relating the vectors b, c
. This is the mixed model which is most commonly used to predict traditional breeding values in animal breeding programmes. For the model using genomic effects, model (1) was changed to a Bayesian multi-marker association model as follows:
fits the genomic effects, with a
the vector representing effects associated with marker alleles
a scaling factor modelling the variance explained by each marker and Q
the design matrix linking alleles with markers [31
]. Priors were assigned to the scaling factor s
as follows for the non-mixture models:
can be interpreted approximately as the expected average fitted variance per marker and TN
denotes a truncated normal distribution. For mixture models the following scaling factors s
where the first distribution models the markers with on average little to no effect at a proportion π
, and the second distribution models the markers that have a substantial effect at a proportion π
. The proportions of markers π
were varied across mixture models ranging from 100 to 2.5%. Variances for the first distribution
were set to 1% of the phenotypic variance of the trait divided by the number of markers. No polygenic effects were present and all other effects were as described for model (1). Using the methodology of genomic selection as described by Meuwissen et al. [1
], it was possible to solve models with more markers than phenotypic records. The last model, which combined both genetics effects of model (1) and (2), can be as described as follows:
where the effects are as defined earlier. Here the polygenic variance of u accounts for genetic variation which could not be explained by the genomic markers a.
PA was calculated as the Pearson’s correlation between a predicted observation and the corresponding realized observation. Realized observation was calculated as the phenotype corrected for fixed effects and covariates, while the predicted observation was the estimated breeding value, as was done by Legarra et al. . To predict these observations, a cross validation approach was used, whereby the dataset was split into a validation set and a training set. The validation set contained the animals for which the observation had to be predicted, while the training set was used to estimate the parameters for the prediction model. Size of the training set is of importance for the estimation of accurate breeding values  and to ensure a sufficient size of training population, a 1:5 proportion of validation to training dataset was used. Only animals from families with at least two members were used to create validation sets (~ 80% of all animals). These animals were randomly split into five groups to create five validation sets. Thus each validation set contained ~16% of all animals. This was repeated to create ten validation sets in total. Each validation set had a corresponding training set, which contained the remaining animals with phenotypic data.
Two different routines for splitting the data were used: within family and between family cross-validation. For within family cross-validation, full sib families were randomly split between training and validation set such that each set contained at least one animal from a family. For between family cross-validation, families were split such that no full sib family would have animals in both datasets simultaneously. As a result, for between family cross-validation no close genetic connectedness due to full sib families was available between training and validation data. In the case of within family cross-validation, full sibs with phenotypic data linked the breeding values of the training and validation data.
Importance of individual markers
As an illustration, the relative importance of individual markers was quantified via the computation of Bayes Factors, conditional on either model (2) or model (3). The correct inferences about the statistical relevance of particular markers could involve, first, calculation of the posterior probability of each model. Secondly one could report Bayes factors conditional on the model with largest posterior probability, or averaging over all models. This task was judged to be computationally too burdensome and was not undertaken in this study. As indicated in Table 7
, traits were chosen across a range of number of QTLs, ranging from as low as 1 for TF and HC up to 20 for W10. The objective was to compare the performance of genomic models (2) and (3) in finding regions with evidence of a marker having an increased effect, and to study how the number of QTLs affecting a trait influences the efficiency of genomic selection. Using the Bayesian approach implemented in the programme iBay [31
], the Bayes Factor computed as the change in prior to posterior odds (PPOR) for each marker was calculated with the following formula:
is the estimate for the posterior probability of the marker having a substantial effect, and π
the a priori probability that the marker has a substantial effect. Results were plotted per trait for all markers, whereby a PPOR > 3.2 can be interpreted as substantial evidence for the marker to have an increased effect, a PPOR > 10 as strong evidence, and a PPOR > 100 as decisive .